For sake of facility in explaining the invention, reference will be had in the following description thereof to a device designed especially for use in connection with bonds and stocks, but it is to be expressly understood that the invention is not thus limited in scope or by the particular device described, it being applicable in its broad principles to many other uses, as will be readily understood by those versed in the art.
The usual question that confronts a prospective buyer of a bond is, what the yield -- net return per period, or yield-to-maturity -- of the bond will be, if purchased at a specified price, and interest or a dividend is paid thereon at the face or coupon rate until maturity (or at any specified time if the value of the bond is known or projected for that time), at which time the bond will be paid at its face or maturity value. (The net return per period or yield-to-maturity is that equivalent rate per period earned on the invested amount which will produce the same total net return as the bond or other security in question).
The manner in which the above desribed factors cooperate to govern the yield rate of a bond is expressed mathematically by the following well known and generally accepted equation: EQU P(1+i).sup.n = C((1+i).sup.n - 1)/i + M (1)
where:
P = purchase or market price of the bond; PA1 C = coupon rate or face interest rate of the bond for the period between coupon payment dates; PA1 n = number of coupon payment periods from date of purchase to date of maturity; PA1 i = net yield rate or yield-to-maturity of the bond if held to maturity or for n periods; and PA1 M = maturity value or value after n periods of the bond. PA1 t = income tax rate, PA1 g = capital gains tax rate, and using equation (1), the following equation, which will be readily understood by those versed in the art, can be developed: EQU Pc1+(1-t)i).sup.n = (1-t)C((1+(1-t)i).sup.n -1)/(1-t)i + (1-g)(M-P) + P. (3) PA1 T = 1 - t, PA1 G = 1 - g, and PA1 I = Ti
The price, therefore, which should be paid for a bond in order to realize a given yield, may be calculated using equation (1). The results of such calculation for certain values of the quantities, within a limited range, have been listed in so-called bond basic books or tables in which the prices, calculated to many decimal places, for different half-year maturities, are listed opposite given yields.
However, the large number of cases which arise in practice are concerned with a reverse operation, namely, determining what the yield will be if the bond is bought at a given market price. Equation (1), however, cannot be readily solved for i, since this variable can be found only by successive approximations (except when the value of n is small or equals infinity) from the following equation, or equivalent representation, derived from equation (1): EQU i = (C((1+i) .sup.n - 1) + iM)/P(1+i).sup.n. C2)
in this formula i is not only present on both sides, but occurs raised to the n.sup.th power, and since the value of n is often very large, according to the number of periods until maturity, this formula is incapable of direct, rapid and accurate solution (except for a few special values of n, or by using sophisticated and expensive electronic computers).
It has, consequently, been customary heretofore to use tables covering many hundreds of pages, calculated in accordance with equation (1), with M = 100. These tables in most cases are used inversely in order to ascertain the yield corresponding to any given market price. However, it would seldom occur that there was listed in the table either the given market price (these prices having been calculated for certain given yields as above described) or the given maturity (since only multiples of half-yearly maturities are shown). It was necessary, therefore, to go through a process of calculation or interpolation to determine the actual yield. These tables, therefore, to be capable of results which are accurate usually require the expenditure of considerable effort and time.
The tables and charts used heretofore have failed to take into account two most significant factors: income tax and capital gains tax. There are many kinds of bonds and investments the returns on which are taxed in different ways making it impossible to use one set of tables or charts which do not consider taxes to cover all the different situations. For example, E-Bonds have no coupon rate but have a maturity value with the return (maturity value minus purchase price) being subject to federal but not state and local income taxes; a Treasury Note or Bond if purchased at a discount will have a return due to the coupon rate which is subject to federal but not state and local income taxes and a capital return (maturity value minus purchase price) which is subject to federal, state and local capital gains tax; certain types of municipal bonds have a coupon return which is subject to no federal income tax and, sometimes, no state or local income tax but the capital return is subject to capital gains tax; corporate bonds are generally subject to both income tax on the coupon return and capital gains tax on the capital return.
The present invention has for one of its objects a device whereby an accurate solution of a problem of the above-described nature may be speedily obtained. The only other means of obtaining accurate solutions is by using sophisticated and expensive electronic computers. The device is so constructed, in fact, that any one of the factors involved may be quickly ascertained if the remaining thereof are known.
By setting
note that if i is the actual return rate, then (1-t)i is the after-tax return; if C is the stated coupon rate, then (1-t)C is the after-tax return; and if (M-P) is the actual capital return, then (1-g) (M-P) is the after-tax return. Also note that when t=0 and g=0 (i.e., there are no taxes), then equation (3) is identical with equation (1).
By setting
equation (3) can be expressed as EQU P(1+I).sup.n = TC((1+I).sup.n - 1)/I + G(M-P) + P (4)
where I is the after-tax yield rate and the other variables are as defined above.